Mistake in Carroll?

On the page 186 of Lecture Notes on General Relativity, where he writes the line element in both ##\tilde{u}## and ##\tilde{v}## coordinates , I think he missed a minus sign. Can someone check it, please?

On the page 186 of Lecture Notes on General Relativity, where he writes the line element in both ##\tilde{u}## and ##\tilde{v}## coordinates , I think he missed a minus sign. Can someone check it, please?

Yes, that's a typo. The first term in equation 7.73 on page 186 should be negative.